Shear StrengthB
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Transcript of Shear StrengthB
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Learning outcomes
Understand the assumptions and limitations of four
soil models: Coulomb, Mohr-Coulomb, Tresca and
Taylor.
Know how to select the appropriate soil model to
interpret soil test data.
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Importance
All models make assumptions. You must understand these
assumptions to know the limitations of a selected model.
The response of soils depends on many factors including the
drainage condition, the history of loading and the stress path.
You must be able to select and use the appropriate model
that best represents the expected soil condition. Poor choice
and use could lead to misrepresentation and failure.
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Key terms
Shear strength of a soil is the maximum internal resistance to applied shearing
forces.
Effective friction angle, , is a measure of the shear strength of soils due to friction.
Cementation, ccm, is a measure of the shear strength of a soil from forces that
cement the particles.
Soil tension, ct, is a measure of the apparent shear strength of a soil from soil suction
(negative pore-water pressures or capillary stresses).
Cohesion, co, is a measure of the intermolecular forces.
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Key terms
Undrained shear strength, su, is the shear strength of a soil when sheared at constant
volume.
Apparent cohesion, C, is the apparent shear strength at zero normal effective stress.
Critical state is a stress state reached in a soil when continuous shearing occurs at
constant shear stress to normal effective stress ratio and constant volume.
Dilation is a measure of the change in volume of a soil when the soil is distorted by
shearing.
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MODELS TO INTERPRET SHEAR STRENGTH
A soil model is an idealized representation of the soil to allow us to understand its response to loading and other external events.
A soil model should not be expected to capture all the intricacies of real soil behavior.
Each soil model may have a different set of assumptions and may only represent one or more aspects of soil behavior.
Popular soil models
Coulomb
Mohr-Coulomb
Tresca
Some other soil models
Taylor
Critical state
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Simple
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COULOMBS SOIL MODEL
Soils, in particular granular soils, are endowed by nature with slip planes.
Each contact of one soil particle with another is a potential micro-slip plane.
Loadings can cause a number of these micro slip planes to align in the direction of least resistance.
Thus, we can speculate that a possible mode of soil failure is slip on a plane of least resistance.
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COULOMBS SOIL MODEL FOR UNCEMENTED,SOILS
LINEAR FAILURE ENVELOPE
Soils at critical state: = cs, = p = 0
CURVED FAILURE ENVELOPE
Soils at peak state: = p, = p > 0
is the dilation angle (a measure of the soils
ability to expand > increase in volume)
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( ) tan ( ) tanf n f p n f cs p
( ) tanf n f cs
cs is a constant for a given soil and is a fundamental soil property; p is not a constant for a given soil it depends on the normal effective stress and the ability of the soil to dilate.
Soil fails by impending frictional sliding on a plane
p cs
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WHAT IS DILATANCY?
Dilation is not a peculiarity of soils,
but occurs in many other materials,
for example, rice and wheat.
The ancient traders of grains were
well aware of the phenomenon of
volume expansion of grains.
However, it was Osborne Reynolds
(1885) who described the
phenomenon of dilatancy and
brought it to the attention of the
scientific community..
Dilation can be seen in action at
a beach.
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COULOMBS SOIL MODEL FOR CEMENTED SOILS
ccm is the cementation strength and o is the apparent friction angle.
Neither ccm nor o is a fundamental soil parameter.
Adding the cementation strength to the apparent frictional strength is not strictly correct since they are not mobilized at the same shear strains.
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( ) tanf cm n f oc
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ISSUES WITH AND USE OF THE COULOMBS MODEL
USE
It can be used for failures that occur along a slip plane, such as a joint or the interface of two soils or the interface between a structure and a soil.
Stratified soil deposits such as overconsolidated varved clays (regular layered soils that depict seasonal variations in deposition) and fissured clays are likely candidates for failure following Coulombs model, especially if the direction of shearing is parallel to the direction of the bedding plane.
ISSUES
Coulombs model applies strictly to two rigid bodies with a common potential sliding plane.
It is a limiting force model (force at impending frictional sliding )
It does not consider soil deformation.
It is independent of the loading history of the soil.
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KEY POINTS REGARDING COULOMBS MODEL
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tan ,f n cs pf
n tan f cm ofc
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MOHRCOULOMB (MC) FAILURE CRITERION
MC failure criterion defines failure when the maximum principal effective stress ratio,
called the maximum effective stress obliquity, is achieved and not when the maximum shear stress
is achieved.
The failure shear stress is then less than the maximum shear stress.
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Soil fails by frictional sliding on a plane of maximum
stress obliquity
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( ),
( )
f
f
1 3[( )/2]max
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MOHRCOULOMB (MC) FAILURE CRITERION
Friction angle
Inclination of failure plane to the
plane of the major principal
stress
Maximum shear stress
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1 3[( )/2]max
1 3
1 3
1 3 1 3
( ) ( )2sin( ) ( )
2
f f
f f
ff f
OB
OA
452 4 2
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MOHRCOULOMB (MC) FAILURE CRITERION
Failure stresses for uncemented
soils
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1 3 1 3( ) sin2 2
n f
1 3 cos2
f
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MC FAILURE CRITERION
Uncemented soils
at critical state
At peak state
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1 3
1 3
sin =cscs
1 3 cos
2cs cs
1 3
1 3
sin =pp
1 3= cos2
p p
p
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MC FAILURE CRITERION
Unsaturated, cemented, cohesive
soils
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o C
Sh
ear
stre
ss
Normal effective
stress, n
1 3
1 3
sin 2 cot +
o
oC
1 31
= tan 1 sin 1 sin2
f o o oC
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ISSUES WITH AND USE OF THE MC MODEL
USE
It can be used for long term
(drained condition) stability
calculations and to interpret
the long term strength of
overconsolidated fine-
grained and dense coarse-
grained soils.
ISSUES
MC model applies strictly to two rigid bodies with a common potential sliding plane.
It is a limiting stress model.
It does not consider soil deformation. Soil deformation is important in real soils.
It is independent of the loading history of the soil. The strength of real soils is dependent on loading history.
The shear strength in compression and extension is the same. Real soils show different strengths in compression and extension. Usually, the extension strength is lower than the compressive strength.
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KEY POINTS: MC FAILURE CRITERION
Coupling Mohrs circle with
Coulombs frictional law allows us
to define shear failure based on
the stress state of the soil.
Failure occurs, according to the
MohrCoulomb failure criterion,
when the soil reaches the
maximum principal effective
stress obliquity.
The maximum shear stress is not
the failure shear stress.
Information on the deformation
or the initial stress state of the
soil is not needed to interpret soil
strength using the MC failure
criterion.
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TRESCAS MODEL
Trescas failure criterion is used to interpret the undrained shear strength.
The shear strength under undrained loading depends only on the initial void ratio or the initial water content or initial confining pressure.
An increase in initial normal effective stress, sometimes called confining pressure, causes a decrease in initial void ratio and a larger change in excess porewater pressure when a soil is sheared under undrained condition.
The result is that the Mohrs circle of total
stress expands and the undrained shear
strength increases. Thus, su is not a
fundamental soil property.
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Soil fails when the shear stress is one-half the principal stress difference
1 3 1 3( ) ( ) ( ) ( )
2 2
f f f f
us
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TRESCAS MODEL
The value of su depends on the
magnitude of the initial confining
pressure or the initial void ratio
(or initial water content).
Analyses of soil strength and soil
stability problems using su are
called total stress analyses (TSA).
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ISSUES WITH AND USE OF THE TRESCAS MODEL
USE
Short term (undrained
condition) stability
calculations and to interpret
the undrained shear strength
of fine-grained soils.
ISSUES
It is a yield criterion for solid bodies that has
been adopted as a failure criterion for soils
(a deformable body).
It is a limiting stress criterion.
It does not consider soil deformation. Soil
deformation is important in real soils.
It is independent of the loading history of the
soil. The strength of real soils is dependent
on loading history.
Compression and expansion strength is the
same. Real soils show different strengths in
compression and in expansion
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KEY POINTS TRESCAS FAILURE CRITERION
For a total stress analysis, which
applies to fine-grained soils, the
shear strength parameter is the
undrained shear strength, su.
Tresca failure criterion is used to
interpret the undrained shear
strength of fine grained soils
The undrained shear strength
depends on the initial void ratio
or initial water content or initial
confining pressure. It is not a
fundamental soil shear strength
parameter.
Information on the deformation
of the soil is not needed to
interpret soil strength using
Tresca failure criterion.
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TAYLORS FAILURE CRITERION
Taylor (1948) used an energy method to derive a simple soil model.
He showed that the shear strength of soil is due to sliding friction from shearing and the interlocking of soil particles.
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The shear strength comes from sliding friction and the interlocking of soil particles
Unlike Coulomb failure criterion, Taylor failure criterion does not require the assumption of any physical mechanism of failure, such as a plane of sliding.
It can be applied at every stage of loading for soils that are homogeneous and deform under plane strain conditions similar to simple shear.
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TAYLORS FAILURE CRITERION: FORMULATION
Equilibrium:
Simplification:
Critical state:
Peak:
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The shear strength comes from sliding friction and the interlocking of soil particles
f z z zd d d
z
f
z
d
d
tan ;f cs 0.zd
d
tan cs
z cs
tan tancs pz p
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ISSUES WITH AND USE OF THE TAYLORS MODEL
USE
Long term stability
calculations of homogeneous
soils.
Cannot be applied to soils
that fail along a joint or an
interface between two soils.
ISSUES
Applies to two-dimensional stress systems.
An extension of Taylor failure criterion
to account for three-dimensional stress is
presented in Chapter 11.
Neither Taylor nor Coulomb failure
criterion explicitly considers the rotation of
the soil particles during shearing.
Gives a higher peak dilation angle than
Coulomb failure criterion.
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DIFFERENCES AMONG THE THREE POPULAR FAILURE CRITERIA
Name Failure criteria Soil treated as Best used for
Test data
interpretation*
Coulomb Failure occurs by
impending, frictional
sliding on a slip plane.
Rigid, frictional
material
Layered or fissured
overconsolidated soils or a
soil where a prefailure
plane exists
Direct shear
Mohr
Coulomb
Failure occurs by
impending, frictional
sliding on the plane of
maximum principal
effective stress obliquity.
Rigid, frictional
material
Long term (drained
condition) strength of
overconsolidated fine-
grained and dense coarse-
grained soils
Triaxial
Tresca Failure occurs when one-
half the maximum
principal stress
difference is achieved.
Homogeneous solid Short term (undrained
condition) strength of fine-
grained soils
Triaxial
* will discuss later
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SUMMARY OF EQUATIONS FOR THE THREE POPULAR FAILURE CRITERIA
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Name Peak Critical state
Coulomb ( ) tan ( ) ( ) tan p n f cs p n f p
unsaturated, cemented soils: ( ) tanf n f oC o t cmC c c c
( ) tan cs n f cs
MohrCoulomb 1 3
1 3
sin pp
1 3
1 3
sin cscs
3 2
1
( ) 1 sintan 45
( ) 1 sin 2
p p p
p p
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1
( ) 1 sintan 45
( ) 1 sin 2
cs cs cs
cs cs
1 3
1 3
Cemented soils: sin 2 cot +
o
oC
o t cmC c c c
Inclination of the failure plane to the plane on which
the major principal effective stress acts.
=45 +2
po
p
Inclination of the failure plane to the plane
on which the major principal effective stress
acts.
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o cs
cs
Tresca
1 3( )
2
p
u ps
1 3( )
2
cs
u css
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RANGES OF FRICTION ANGLES AND DILATION ANGLES FOR SOILS
Ranges of Friction Angles for Soils (degrees)
Soil type
Gravel 3035 3550
Mixtures of gravel and
sand with fine-grained
soils 2833 3040
Sand 2737* 3250
Silt or silty sand 2432 2735
Clays 1530 2030 515
*Higher values (3237) in the range are for sands
with significant amount of feldspar (Bolton, 1986).
Lower values (2732) in the range are for quartz
sands.
Typical Ranges of Dilation Angles for Soils
Soil type
(degrees)
Dense sand 1015
Loose sand
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TYPICAL VALUES OF SU FOR SATURATED FINE-GRAINED SOILS
Description su (kPa( su (psf)
Very soft < 10 200 > 4000
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Quiz 1 30
Which failure criterion (model) is best suited to analyze
the potential failure of the soil mass shown?
1. Mohr-Coulomb
2. Coulomb
3. Tresca
4. None of the above
Dense sand
Stiff overconsolidated clay
Loose silty sand
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Quiz 2 31
The critical state friction angle of a soil is 30 degrees.
If the normal effective stress imposed by a building is
100 kPa, the shear stress (kPa) to cause failure is most
nearly
1. 86.6
2. 100
3. 50
4. 57.7
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Quiz 3 32
The critical state friction angle of a soil is 30 degrees.
The ratio of the major principal effective stress to the
minor principal effective stress to cause failure is most
nearly
1. 0.5
2. 1
3. 2
4. 3
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PRACTICAL IMPLICATIONS OF FAILURE CRITERIA
Region I. Impossible soil
states. A soil cannot have soil
states above the boundary
AEFB.
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PRACTICAL IMPLICATIONS OF FAILURE CRITERIA
Region II. Impending instability (risky
design).
Region AEFA is characteristic of dilating
soils that show peak shear strength and
are associated with the formation of
shear bands. The shear bands consist of
soils that have reached the critical state
and are embedded within soil zones with
high interlocking stresses due to particle
rearrangement. These shear bands grow
as the peak shear strength is mobilized
and as the soil strain-softens subsequent
to the critical state.
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PRACTICAL IMPLICATIONS OF FAILURE CRITERIA
Region III. Stable soil states (safe design).
One of your aims as a geotechnical engineer is to design geotechnical systems on the basis that if the failure state were to occur, the soil would not collapse suddenly but would continuously deform under constant load. This is called ductility. Soil states that are below the failure line or failure envelope AB would lead to safe design. Soil states on AB are failure (critical) states
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KEY POINTS
There are several failure criteria for soils.
Each criterion has application to certain soil conditions.
The three popular failure criteria (Coulomb, Mohr-Coulomb and Tresca) assume that soil is a rigid-plastic material with no deformation prior to failure.
The Coulomb and Mohr-Coulomb failure criteria are applicable to estimate long term failure.
The Mohr-Coulomb failure criterion also assume that failure shear strength of soil in compression and extension is the same. In reality, the shear strength at failure in extension is less than in compression.
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KEY POINTS
Soil states above the peak shear strength boundary are impossible.
Soil states within the peak shear strength boundary and the failure line (critical state) are associ-ated with brittle, discontinuous soil responses and risky design.
Soil states below the failure line lead to ductile responses and are safe.
You should not rely on p in geotechnical design, because the amount of dilation one measures in laboratory or field tests may not be mobilized by the soil under construction loads. You should use cs unless experience dictates otherwise.
A higher factor of safety is warranted if p rather than cs is used in design.
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KEY POINTS
The undrained shear strength, su, applies only to fine-grained soils.
The undrained shear strength is not a fundamental soil parameter.
The undrained shear strength depends on the initial void ratio or initial confining pressure (consolidation pressure).
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